Bianchini and Scarselli [1]}, [2]} design a topological measure of model complexity for deep neural networks. Given an FCNN with single output, denoted by \(\mathcal {N}: \mathbb {R}^d \rightarrow \mathbb {R}\) , they define \(S=\lbrace x \in \mathbb {R}^d | \mathcal {N}(x) \ge 0\rbrace \)  the set of instances being classified by \(\mathcal {N}\)  to the positive class. The model complexity of neural network \(\mathcal {N}\)  is measured by \(B(S)\) , the sum of the Betti numbers of set \(S\) , that is,
\(\nonumber MEC(\mathcal {N}) = B(S) = \sum _{i=0}^{d-1} b_i(S)\) 
